Why checking is not proving, why one counterexample is enough, and how to lay out a proof that earns the marks.
Euler noticed that n² + n + 41 seems to give a prime every time. Test it, one value at a time.
So far, so convincing.
Forty correct cases proved nothing. At n = 40 the formula gives 40² + 40 + 41 = 1681, which is 41 × 41. You can see it without any arithmetic once it is pointed out: put n = 41 and every term has a factor of 41.
Checking cases can never prove a general claim, because there are infinitely many cases and you will always have tested finitely many.
But one counterexample disproves one, completely and for ever. The two are not symmetric, and that asymmetry is the whole reason proof exists.
The examined format is left-hand side to right-hand side. Start with one side, transform it, arrive at the other. Never work on both sides at once, and never start by assuming what you are proving.
The commonest way to lose every mark is to begin by writing the thing you are proving and then rearranging it to 0 = 0. That assumes the result. Start from one side only.
1. Evaluate n² + n + 41 at n = 40.
2. How many counterexamples are needed to disprove “n² + n + 41 is always prime”?
3. A student proves an identity by writing it down and rearranging until both sides read 0 = 0. This:
Starting from one side and never touching the other until you arrive at it.
Every line following from the one above by a stated step, not a leap.
Ending with a clear statement that the two sides are equal. A proof that trails off has not finished.
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